Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations

TL;DR

This paper introduces a physics-guided data augmentation method to enhance neural operator models for solving linear differential equations, reducing data requirements and improving robustness.

Contribution

The proposed PGDA method leverages physical properties of differential equations to augment training data, improving neural operator accuracy and generalization.

Findings

PGDA improves neural operator performance on linear differential equations.

It reduces the amount of ground truth data needed for training.

PGDA enhances robustness to distributional shifts.

Abstract

Neural networks, especially the recent proposed neural operator models, are increasingly being used to find the solution operator of differential equations. Compared to traditional numerical solvers, they are much faster and more efficient in practical applications. However, one critical issue is that training neural operator models require large amount of ground truth data, which usually comes from the slow numerical solvers. In this paper, we propose a physics-guided data augmentation (PGDA) method to improve the accuracy and generalization of neural operator models. Training data is augmented naturally through the physical properties of differential equations such as linearity and translation. We demonstrate the advantage of PGDA on a variety of linear differential equations, showing that PGDA can improve the sample complexity and is robust to distributional shift.